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IEEE International Conference on Computational Electromagnetics

The Maxwell-Schrödinger Model for Three-Level Atomic System with Electromagnetic Field

Junbo Liu, Jun Hu, Yongpin Chen

Published Mar 1, 2018• 1 citation
Publisher

Abstract

In lasing without inversion, the essential idea is the absorption in cancellation by atomic coherence and interference. Usually this is accomplished in three-level atomic systems. In this work, the interaction of a three-level atom and the external electromagnetic field is discussed by the Maxwell-Schrddinger model. The atom is treated quantum mechanically by the Schrddinger equation, whereas the field is treated classically by the Maxwell's equations. An example is presented to demonstrate the effect of coherent trapping.

DOI
10.1109/compem.2018.8496497

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In lasing without inversion, the essential idea is the absorption in cancellation by atomic coherence and interference. Usually this is accomplished in three-level atomic systems. In this work, the interaction of a three-level atom and the external electromagnetic field is discussed by the Maxwell-Schrddinger model. The atom is treated quantum mechanically by the Schrddinger equation, whereas the field is treated classically by the Maxwell's equations. An example is presented to demonstrate the effect of coherent trapping.
A full quantum treatment is given of suppression of absorption and lasing without inversion in a ∧-type three-level atom induced by a strong driving field resonant with the two lower levels. The nature of the quantum interference leading to the suppression of absorption and lasing without inversion and its relation to the initial state preparation are studied. Particular attention is given to the effects that arise specifically from the quantum nature of the driving field. It is shown that, because of the distinguishability of the states of the driving field that have different photon numbers, destructive interference is confined to occur within a finite region of the photon number space. A complete cancellation of absorption predicted in the semiclassical theory should thus be regarded as resulting from an ideal behavior achieved only in the classical limit when the driving field is given as a strong coherent field.
This chapter introduces a three-level atomic model and discusses two other coherent atomic phenomena; namely, coherent Raman scattering and lasing without inversion. Lasing without inversion may take place in three-level resonant atomic systems and, like super-radiance, results from atomic coherence. Super-radiation is initiated by spontaneous fluctuations of the polarization in the medium. The semiclassical method is a powerful tool that is useful for the treatment of propagation effects in super-radiance as well as in such coherent emission processes as self-induced transparency, photon echo and others. The use of the Maxwell—Bloch equations allows one to consider these phenomena in elongated systems. Fortunately, the semiclassical approach is readily applicable to low-density systems that are to gases. However, one has to exercise greater care when applying it to coherent emission in solids. In high-density systems there is a strong mixing of the field and atomic states in the vicinity of resonance.
Quantum coherence and correlations in atomic and radiation physics have led to many interesting and unexpected consequences. For example, an atomic ensemble prepared in a coherent superposition of states yields the Hanle effect, quantum beats, photon echo, self-induced transparency, and coherent Raman beats. In fact, in Section 1.4, we saw that the quantum beat effect provides one of the most compelling reasons for quantizing the radiation field. A further interesting consequence of preparing an atomic system in a coherent superposition of states is that, under certain conditions, it is possible for atomic coherence to cancel absorption. Such atomic states are called trapping states †. The observation of nonabsorbing resonances via atomic coherence and interference impacts on the concepts of lasing without inversion (LWI),‡ enhancement of the index of refraction accompanied by vanishing absorption, and electromagnetically induced transparency. In lasing without inversion, the essential idea is the absorption cancellation by atomic coherence and interference. This phenomenon is also the essence of electromagnetically induced transparency. Usually this is accomplished in three-level atomic systems in which there are two coherent routes for absorption that can destructively interfere, thus leading to the cancellation of absorption. A small population in the excited state can thus lead to net gain. A related phenomenon is that of resonantly enhanced refractive index without absorption in an ensemble of phase-coherent atoms ( phaseonium ). In a phaseonium gas with no population in the excited level, the absorption cancellation always coincides with vanishing refractivity.
The LWI operation condition for an open Λ-type three level atomic system is obtained by solving the steady-state density-matrix equation with the system being driven by a strong coherent field while the inherent pumping rate and decay rate being arbitrary.It is shown that the inversions of inherent pumping rate and spontaneous decay rate replace the inversion of population,and together help to realize lasing without population inversion.Furthermore,the LWI can be realized if just either one of the two inversions is fulfilled.A schrodinger equation of three-level atomic system with appropriate conditions is analytically solved to explain the quantum interference process that leads to LWI.
We study the Maxwell–Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves 6 + N(N + 1)/2 unknowns, the six-dimensional electromagnetic field plus the N(N + 1)/2 unknowns describing the state of matter, where N is the number of atomic energy levels of the considered material. We consider at once a high frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrödinger-rate model, involving 3 + N unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in ℂ3. It satisfies a Schrödinger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a N-dimensional vector describing the occupation numbers of each atomic level. It satisfies Einstein's rate equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two-dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an ad hoc Ansatz.